Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#33703 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,5)(4,6)(7,8)(9,10,16,12)(11,22,24,36)(13,29,18,20)(14,28,32,19)(15,30,31,21)(17,33,23,25)(26,35,39,37)(27,34,40,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3,4,7)(2,5,6,8)(9,11,14,26,15,27,13,25,16,24,32,39,31,40,18,33)(10,17,20,34,21,35,19,22,12,23,29,38,30,37,28,36);(2,6)(5,8)(9,12,13,28,31,21,14,29,16,10,18,19,15,30,32,20)(11,23,25,36,40,35,26,38,24,17,33,22,27,37,39,34);(11,24)(17,23)(22,36)(25,33)(26,39)(27,40)(34,38)(35,37)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26691);